Corner detection: the basic idea
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1 Corner detection: the basic idea At a corner, shifting a window in any direction should give a large change in intensity flat region: no change in all directions edge : no change along the edge direction corner : significant change in all directions 6
2 A simple corner detector Define the sum squared difference (SSD) between an image patch and a patch shifted by offset (x,y): where w(u,v) = or 1 in window, 0 outside Gaussian If s(x,y) is high for shifts in all 8 directions, declare a corner. Problem: not isotropic 7
3 Harris corner detector derivation Second-order Taylor series approximation: where A is defined in terms of partial derivatives I x = I/ x and I y = I/ y summed over (u,v): For constant t, S(x,y) < t is an ellipse 8
4 Eigenvector analysis The eigenvectors v 1, v 2 of A give an orthogonal basis for the ellipse I.e. directions of fastest and slowest change for λ 2 > λ 1, v 1 is the direction of fastest change (minor axis of ellipse) and v 2 is the direction of slowest change (major axis) direction of the fastest change v 2 direction of the slowest change (λ 2 ) -1/2 (λ 1 ) -1/2 v 1 9
5 Classify points based on eigenvalues Classification of image points using eigenvalues of M: λ 2 Edge λ 2 >> λ 1 Corner λ 1 and λ 2 are large, λ 1 ~ λ 2 ; E increases in all directions λ 1 and λ 2 are small; E is almost constant in all directions Flat region Edge λ 1 >> λ 2 λ 1 10
6 Harris corner detection But square roots are expensive Approximate corner response function that avoids square roots: R ( ) 2 = λ λ k λ + λ with k is set empirically After thresholding, keep only local maxima of R as corners prevents multiple detections of the same corner 13
7 Harris detector, step-by-step 14
8 Harris detector, step-by-step Compute corner response R 15
9 Harris detector, step-by-step Threshold on corner response R 16
10 Harris detector, step-by-step Take only local maxima of R 17
11 Harris detector properties Invariant to intensity shift: I = I + b only derivatives are used, not original intensity values Insensitive to intensity scaling: I = a I R R threshold x (image coordinate) x (image coordinate) So Harris is insensitive to affine intensity changes I.e. linear scaling plus a constant offset, I = a I + b 21
12 Rotation invariance Harris detector properties Ellipse (eigenvectors) rotate but shape (eigenvalues) remain the same Corner response R is invariant to image rotation 20
13 Harris detector properties But Harris is not invariant to image scale All points will be classified as edges Corner! 22
14 Experimental evaluation Quality of Harris detector for different scale changes Repeatability rate: # correspondences # possible correspondences C.Schmid et.al. Evaluation of Interest Point Detectors. IJCV
15 Scale invariant interest point detection Consider regions (e.g. circles) of different sizes around a point Regions of corresponding sizes will look the same in both images 25
16 Scale invariant detection The problem: how do we choose corresponding circles independently in each image? 26
17 A solution Design a function which is scale invariant I.e. value is the same for two corresponding regions, even if they are at different scales Example: average intensity is the same for corresponding regions, even of different sizes For a given point in an image, consider the value of f as a function of region size (circle radius) f scale = 1/2 f region size region 27 size
18 A solution Take a local maximum of this function The region size at which maximum is achieved should be invariant to image scale This scale invariant region size is determined independently in each image f scale = 1/2 f s 1 region size s 2 region 28 size
19 Choosing a function A good function for scale detection has one sharp peak f f f Good! bad bad region size region size region size A function that responds to image contrast is a good choice e.g. convolve with a kernel like the Laplacian or the Difference of Gaussians 29
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